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Date: 1-9-2019
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Date: 9-10-2019
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Date: 25-3-2019
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The polynomials given by the associated Sheffer sequence with
(1) |
where . The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function
(2) |
can be given in terms of the sum
(3) |
This results in
(4) |
where is a falling factorial. The first few are
(5) |
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(6) |
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(7) |
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(8) |
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(9) |
The binomial identity obtained from the Sheffer sequence gives the generalized Chu-Vandermonde identity
(10) |
(Roman 1984, p. 69; typo corrected).
In the special case , the function simplifies to
(11) |
which gives the generating function
(12) |
giving the polynomials
(13) |
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(14) |
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(15) |
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(16) |
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(17) |
REFERENCES:
Gould, H. W. "Note on a Paper of Sparre-Anderson." Math. Scand. 6, 226-230, 1958.
Gould, H. W. "Stirling Number Representation Problems." Proc. Amer. Math. Soc. 11, 447-451, 1960.
Gould, H. W. "A Series of Transformation for Finding Convolution Identities." Duke Math. J. 28, 193-202, 1961.
Gould, H. W. "Note on a Paper of Klamkin Concerning Stirling Numbers." Amer. Math. Monthly 68, 477-479, 1961.
Gould, H. W. "A New Convolution Formula and Some New Orthogonal Relations for the Inversion of Series." Duke Math. J. 29, 393-404, 1962.
Gould, H. W. "Congruences Involving Sums of Binomial Coefficients and a Formula of Jensen." Amer. Math. Monthly 69, 400-402, 1962.
Roman, S. "The Gould Polynomials and he Central Factorial Polynomials." §4.1.4 in The Umbral Calculus. New York: Academic Press, pp. 67-70, 1984.
Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.
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دراسة: عدم ترتيب الغرفة قد يدل على مشاكل نفسية
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علماء: تغير المناخ تسبب في ارتفاع الحرارة خلال موسم الحج
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جهود مكثفة وطباعة عشرات الآلاف من المنشورات .. استعدادات العتبة العلوية المقدسة لعيد الغدير الأغر
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